Matrix F-distribution

In statistics, the matrix F distribution (or matrix variate F distribution) is a matrix variate generalization of the F distribution which is defined on real-valued positive-definite matrices. In Bayesian statistics it can be used as the semi conjugate prior for the covariance matrix or precision matrix of multivariate normal distributions, and related distributions.

Density
The probability density function of the matrix $$F$$ distribution is:

$$ f_{\mathbf X}({\mathbf X}; {\mathbf \Psi}, \nu, \delta) = \frac{\Gamma_p\left(\frac{\nu+\delta+p-1}{2}\right)}{\Gamma_p\left(\frac{\nu}{2}\right)\Gamma_p\left(\frac{\delta+p-1}{2}\right)|\mathbf{\Psi}|^{\frac{\nu}{2}}}~|{\mathbf X}|^{\frac{\nu-p-1}{2}} |\textbf{I}_p+{\mathbf X}\mathbf{\Psi}^{-1}|^{-\frac{\nu+\delta+p-1}{2}} $$

where $$\mathbf{X}$$ and $${\mathbf\Psi}$$ are $$p\times p$$ positive definite matrices, $$| \cdot |$$ is the determinant, &Gamma;p(&sdot;) is the multivariate gamma function, and $$\textbf{I}_p$$ is the p × p identity matrix.

Construction of the distribution
$${\mathbf \Phi_1}\sim \mathcal{W}({\mathbf I_p},\nu)$$ and $${\mathbf \Phi_2}\sim \mathcal{W}({\mathbf I_p},\delta+k-1)$$, and define $$\mathbf X = {\mathbf \Phi_2}^{-1/2}{\mathbf \Phi_1}{\mathbf \Phi_2}^{-1/2}$$, then $$\mathbf X\sim \mathcal{F}({\mathbf I_p},\nu,\delta) $$.
 * The standard matrix F distribution, with an identity scale matrix $$\mathbf I_p$$, was originally derived by. When considering independent distributions,

$$ f_{\mathbf X | \mathbf\Phi, \nu, \delta}(\mathbf X) = \int f_{\mathbf X | \mathbf\Phi, \delta+p-1}(\mathbf X) f_{\mathbf\Phi | \mathbf\Psi, \nu}(\mathbf\Phi) d\mathbf\Phi. $$ This construction is useful to construct a semi-conjugate prior for a covariance matrix.
 * If $${\mathbf X}|\mathbf\Phi\sim \mathcal{W}^{-1}({\mathbf\Phi},\delta+p-1)$$ and $${\mathbf \Phi}\sim \mathcal{W}({\mathbf\Psi},\nu)$$, then, after integrating out $$\mathbf\Phi$$, $$\mathbf X$$ has a matrix F-distribution, i.e.,

f_{\mathbf X | \mathbf\Psi, \nu, \delta}(\mathbf X) = \int f_{\mathbf X | \mathbf\Phi, \nu}(\mathbf X) f_{\mathbf\Phi | \mathbf\Psi, \delta + p - 1}(\mathbf\Phi) d\mathbf\Phi. $$ This construction is useful to construct a semi-conjugate prior for a precision matrix.
 * If $${\mathbf X}|\mathbf\Phi\sim \mathcal{W}({\mathbf\Phi},\nu)$$ and $${\mathbf \Phi}\sim \mathcal{W}^{-1}({\mathbf\Psi},\delta+p-1)$$, then, after integrating out $$\mathbf\Phi$$, $$\mathbf X$$ has a matrix F-distribution, i.e., $$

Marginal distributions from a matrix F distributed matrix
Suppose $${\mathbf A}\sim F({\mathbf\Psi},\nu,\delta)$$ has a matrix F distribution. Partition the matrices $$ {\mathbf A} $$ and $$ {\mathbf\Psi} $$ conformably with each other

{\mathbf{A}} = \begin{bmatrix} \mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{21} & \mathbf{A}_{22} \end{bmatrix}, \; {\mathbf{\Psi}} = \begin{bmatrix} \mathbf{\Psi}_{11} & \mathbf{\Psi}_{12} \\ \mathbf{\Psi}_{21} & \mathbf{\Psi}_{22} \end{bmatrix} $$ where $${\mathbf A_{ij}}$$ and $${\mathbf \Psi_{ij}} $$ are $$ p_{i}\times p_{j}$$ matrices, then we have $$ {\mathbf A_{11} } \sim F({\mathbf \Psi_{11} }, \nu, \delta) $$.

Moments
Let $$ X \sim F({\mathbf\Psi},\nu,\delta)$$.

The mean is given by: $$ E(\mathbf X) = \frac{\nu}{\delta-2}\mathbf\Psi.$$

The (co)variance of elements of $$\mathbf{X}$$ are given by:



\operatorname{cov}(X_{ij},X_{ml}) = \Psi_{ij}\Psi_{ml}\tfrac{2\nu^2+2\nu(\delta-2)}{(\delta-1)(\delta-2)^2(\delta-4)} + (\Psi_{il}\Psi_{jm}+\Psi_{im}\Psi_{jl})\left(\tfrac{2\nu+\nu^2(\delta-2)+\nu(\delta-2)}{(\delta-1)(\delta-2)^2(\delta-4)}+\tfrac{\nu}{(\delta-2)^2}\right). $$

Related distributions
f_{x\mid\nu, \delta}(x) = \operatorname{B}\left(\tfrac{\nu}{2},\tfrac{\delta}{2}\right)^{-1} \left(\tfrac{\nu}{\delta}\right)^{\nu/2} x^{\nu/2 - 1} \left(1+\tfrac{\nu}{\delta} \, x \right)^{-(\nu+\delta)/2}, $$
 * The matrix F-distribution has also been termed the multivariate beta II distribution. See also, for a univariate version.
 * A univariate version of the matrix F distribution is the F-distribution. With $$p=1$$ (i.e. univariate) and $$\mathbf\Psi = 1$$, and $$x=\mathbf{X}$$, the probability density function of the matrix F distribution becomes the univariate (unscaled) F distribution: $$


 * In the univariate case, with $$p=1$$ and $$x=\mathbf{X}$$, and when setting $$\nu=1$$, then $$\sqrt{x}$$ follows a half t distribution with scale parameter $$\sqrt{\psi}$$ and degrees of freedom $$\delta$$. The half t distribution is a common prior for standard deviations